I'm still unsatisfied with your definitions because you're not separating the concept of the expression of a computation from the execution of that computation. A formalized process (read: language) is Turing complete if it can express the simulation of a Turing machine. Since we know a Turing machine can be used to compute anything that is computable, we then know our formalized process can compute all that is compute - it is Turing Complete.

Programs cannot be Turing complete. The concept does not apply, just as a basketball cannot be sad. Programming languages can be Turing complete. A robotic Lego "Turing machine" is not Turing complete, but not because it's not a realization of a Turing machine, but because it's not a formalized process. Robotic Legos, however, would be Turing complete, since that is the formal process used to simulate the Turing machine.